Answer :
To multiply and simplify the product [tex]\((8 - 5i)^2\)[/tex], we can follow these steps:
1. Understand the Expression: We are given the expression [tex]\((8 - 5i)^2\)[/tex]. This means we need to square the complex number [tex]\(8 - 5i\)[/tex].
2. Apply the Formula: The square of a complex number [tex]\((a + bi)\)[/tex] is given by:
[tex]\[
(a + bi)^2 = a^2 + 2abi + (bi)^2
\][/tex]
3. Identify the Values: Here, [tex]\(a = 8\)[/tex] and [tex]\(b = -5\)[/tex]. So the expression is [tex]\((8 + (-5)i)^2\)[/tex].
4. Calculate Each Term:
- Calculate [tex]\(a^2\)[/tex]: [tex]\(8^2 = 64\)[/tex].
- Calculate [tex]\(2abi\)[/tex]: [tex]\(2 \times 8 \times (-5)i = -80i\)[/tex].
- Calculate [tex]\((bi)^2\)[/tex]: [tex]\((-5i)^2 = 25i^2\)[/tex], and since [tex]\(i^2 = -1\)[/tex], this becomes [tex]\(25 \times (-1) = -25\)[/tex].
5. Combine the Results:
- The real part is: [tex]\(64 - 25 = 39\)[/tex].
- The imaginary part is: [tex]\(-80i\)[/tex].
6. Write the Simplified Form: The final answer is:
[tex]\[
39 - 80i
\][/tex]
Based on this step-by-step calculation, the product of [tex]\((8 - 5i)^2\)[/tex] is [tex]\(39 - 80i\)[/tex]. Therefore, the correct answer from the given options is [tex]\(\boxed{39 - 80i}\)[/tex].
1. Understand the Expression: We are given the expression [tex]\((8 - 5i)^2\)[/tex]. This means we need to square the complex number [tex]\(8 - 5i\)[/tex].
2. Apply the Formula: The square of a complex number [tex]\((a + bi)\)[/tex] is given by:
[tex]\[
(a + bi)^2 = a^2 + 2abi + (bi)^2
\][/tex]
3. Identify the Values: Here, [tex]\(a = 8\)[/tex] and [tex]\(b = -5\)[/tex]. So the expression is [tex]\((8 + (-5)i)^2\)[/tex].
4. Calculate Each Term:
- Calculate [tex]\(a^2\)[/tex]: [tex]\(8^2 = 64\)[/tex].
- Calculate [tex]\(2abi\)[/tex]: [tex]\(2 \times 8 \times (-5)i = -80i\)[/tex].
- Calculate [tex]\((bi)^2\)[/tex]: [tex]\((-5i)^2 = 25i^2\)[/tex], and since [tex]\(i^2 = -1\)[/tex], this becomes [tex]\(25 \times (-1) = -25\)[/tex].
5. Combine the Results:
- The real part is: [tex]\(64 - 25 = 39\)[/tex].
- The imaginary part is: [tex]\(-80i\)[/tex].
6. Write the Simplified Form: The final answer is:
[tex]\[
39 - 80i
\][/tex]
Based on this step-by-step calculation, the product of [tex]\((8 - 5i)^2\)[/tex] is [tex]\(39 - 80i\)[/tex]. Therefore, the correct answer from the given options is [tex]\(\boxed{39 - 80i}\)[/tex].
